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Discrete Poincare and Bogovskii operators on cochains and Whitney forms

Johnny Guzmán, Anil N. Hirani, Bingyan Liu, Pratyush Potu

Abstract

Smooth Poincare operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We construct analogous discrete Poincare operators acting on cochains and Whitney forms. We provide explicit, constructive realizations of these operators under various assumptions on the underlying domain or simplicial complex. In particular, we provide simple constructions for the discrete Poincare operators on simplicial complexes which are collapsible and those with underlying domain being star-shaped with respect to a point. We then provide more abstract constructions on simplicial complexes which are discrete contractible and domains which are Lipschitz contractible. We also modify the discrete Poincare operator on star-shaped domains to construct a discrete Bogovskii operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC).

Discrete Poincare and Bogovskii operators on cochains and Whitney forms

Abstract

Smooth Poincare operators are a tool used to show the vanishing of smooth de Rham cohomology on contractible manifolds and have found use in the analysis of finite element methods based on the Finite Element Exterior Calculus (FEEC). We construct analogous discrete Poincare operators acting on cochains and Whitney forms. We provide explicit, constructive realizations of these operators under various assumptions on the underlying domain or simplicial complex. In particular, we provide simple constructions for the discrete Poincare operators on simplicial complexes which are collapsible and those with underlying domain being star-shaped with respect to a point. We then provide more abstract constructions on simplicial complexes which are discrete contractible and domains which are Lipschitz contractible. We also modify the discrete Poincare operator on star-shaped domains to construct a discrete Bogovskii operator which satisfies the requisite homotopy identity while preserving homogeneous boundary conditions. Applications arise in the construction of discrete scalar and vector potentials and in the discrete wedge product of Discrete Exterior Calculus (DEC).

Paper Structure

This paper contains 31 sections, 21 theorems, 95 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smooth Poincaré operators
  4. Singular simplices and chains
  5. Simplicial and singular chain maps
  6. Discrete Poincaré operators
  7. Generalized cone operators
  8. Discrete Poincaré operators
  9. Some simple examples of cone operators
  10. Simplicial cone operator on collapsible simplicial complexes
  11. Singular cone operator on star-shaped domains
  12. Cone operators more generally
  13. Product Space Triangulations and Extrusions
  14. Simplicial cone operator on discrete contractible simplicial complexes
  15. Singular cone operator on Lipschitz contractible domains
  16. ...and 16 more sections

Key Result

Proposition 3.5

Let $X$ be a simplicial complex and ${\mathop{\textup{\bf Co}}}_k$ be a simplicial cone operator with contraction vertex $a$. Let the associated discrete Poincaré operator of Definition def:comb_disc_Poinc be denoted $P^k$. Then, it holds where $\pi_{\mathcal{C}}^* \alpha$ is such that $\bigl\langle \pi_{\mathcal{C}}^* \alpha, v\bigr\rangle = \bigl\langle \alpha, a\bigr\rangle$ for any $v\in X^0$

Figures (9)

  • Figure 1: A sequence of five elementary simplicial collapses reducing a complex of two adjacent triangles to a single vertex $a$.
  • Figure 2: Illustration of different scenarios for the region $[a,\sigma]$.
  • Figure 3: Different admissible product complexes of $\Omega \times I$ where the complex $X$ consists of two edges $\sigma_1$ and $\sigma_2$ which meet at a vertex.
  • Figure 4: A strong collapse sequence $X=X_4\mathrel{\searrow\mkern-8mu\searrow} X_3 \mathrel{\searrow\mkern-8mu\searrow} \cdots \mathrel{\searrow\mkern-8mu\searrow} X_0 = a$. Because $a$ belongs to every maximal simplex, it dominates all other vertices. In the first step $X_4 \mathrel{\searrow\mkern-8mu\searrow} X_3$, removing the dominated vertex $v_2$ simultaneously removes two $2$-simplices. Moreover at each strong collapse step, a vertex is removed. Both of these facts are in contrast to an elementary simplicial collapse.
  • Figure 5: Examples of different scenarios for the images of infinite singular cones.
  • ...and 4 more figures

Theorems & Definitions (81)

  • Definition 2.1: Singular simplices
  • Definition 2.2: Singular chains
  • Definition 2.3: Linear singular simplices
  • Definition 2.4: Lipschitz singular simplices
  • Remark 2.5: Degenerate simplices
  • Definition 2.6: Simplicial map
  • Definition 2.7: Induced simplicial chain map
  • Definition 2.8: Induced singular chain map
  • Remark 2.9
  • Remark 3.1
  • ...and 71 more